One limited study of monozygotic and dizygotic twins claims that genetic variation can have an effect on reactions to unfair offers, though the study failed to employ actual controls for environmental differences.
When carried out between members of a shared social group (e.g., a village, a tribe, a nation, humanity) people offer "fair" (i.e., 50:50) splits, and offers of less than 30% are often rejected. The first experimental analysis of the ultimatum game was by Werner Güth, Rolf Schmittberger, and Bernd Schwarze: Their experiments were widely imitated in a variety of settings.
The weak equilibrium is an artifact of the strategy space being continuous. No share with S > 0 is subgame perfect, because the proposer would deviate to S' = S - ϵ and the receiver's best response would still be to accept. It is weak because the receiver's payoff is 0 whether he accepts or rejects. The unique subgame perfect equilibrium is ( S=0, Accept). If the receiver rejects the offer, both players get zero. If the receiver accepts the offer, the proposer's payoff is (1-S) and the receiver's is S. Suppose the proposer chooses a share S of a pie to offer the receiver, where S can be any real number between 0 and 1, inclusive. The ultimatum game is also often modelled using a continuous strategy set. This would have two subgame perfect equilibria: (Proposer: S=0, Accepter: Accept), which is a weak equilibrium because the acceptor would be indifferent between his two possible strategies and the strong (Proposer: S=1, Accepter: Accept if S>=1 and Reject if S=0). For example, the item being shared might be a dollar bill, worth 100 cents, in which case the proposer's strategy set would be all integers between 0 and 100, inclusive for his choice of offer, S. A more realistic version would allow for many possible offers. The simplest version of the ultimatum game has two possible strategies for the proposer, Fair and Unfair. So, the first two Nash equilibria above are not subgame perfect: the responder can choose a better strategy for one of the subgames. In both subgames, it benefits the responder to accept the offer. The theory relies on the assumption that players are rational and utility maximising. A perfect-subgame equilibrium occurs when there are Nash Equilibria in every subgame, that players have no incentive to deviate from. The above game can be viewed as having two subgames: the subgame where the proposer makes a fair offer, and the subgame where the proposer makes an unfair offer. However, only the last Nash equilibrium satisfies a more restrictive equilibrium concept, subgame perfection. The proposer makes an unfair offer the responder would accept any offer.The proposer makes an unfair offer the responder would only accept an unfair offer.The proposer makes a fair offer the responder would only accept a fair offer.So, there are three Nash equilibria for this game: Meanwhile, it benefits the proposer to make an offer that the responder will accept furthermore, if the responder would accept any offer, then it benefits the proposer to switch from a fair to an unfair offer.
It always benefits the responder to accept the offer, as receiving something is better than receiving nothing. For each of these two splits, the responder can choose to accept or reject, which means that there are four strategies available to the responder: always accept, always reject, accept only a fair split, or accept only an unfair split.Ī Nash equilibrium is a pair of strategies (one for the proposer and one for the responder), where neither party can improve their reward by changing strategy. There are two strategies available to the proposer: propose a fair split, or propose an unfair split. The argument given in this section can be extended to the more general case where the proposer can choose from many different splits.